I am trying to find a bound for these Salie-type sums. When $\beta = 1$, the bound $|S(a,b,1)| \leq 2 \sqrt{p}$ is due to Salie. When $\beta \geq 2$ is even, the Jacobi symbol is identically $1$ and so the sum reduces to a Kloosterman sum

2 Answers
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There is a general "elementary" formula for Salié sums for arbitrary modulus, involving roots of quadratic equations, and from which the bound is immediate. A quick derivation is in Sarnak's "Some applications of modular forms" but it can be found in many places.

Another nice reference is H. Iwaniec's Topics in classical automorphic forms, in which he computed the classical Kloosterman sums and Salie sums in an elementary manner for the prime power moduli.
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arithboyMay 19 '11 at 6:44

In fact, a similar upper bound also holds if the Jacobi symbol is replaced by any other Dirichlet character mod $p^\beta$.
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arithboyMay 19 '11 at 6:46

Why did this get voted up? That paper does not address the question that is asked, as it only treats the case of sums over finite fields, not finite rings as in the question (with beta > 1). –
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KConradMay 19 '11 at 6:51